West Ham managed a consolation 11 minutes from time as Antonio fired through the legs of Fernandez and past the unsighted Fabianski, but the storm clouds continue to gather.
West Ham United manager David Moyes has warned his team to expect a hard test when they travel to face a resurgent Swansea City in Saturday's Premier League clash as both teams fight to avoid relegation.
Adverse weather had caused the West Ham flight to be cancelled and there were more problems for the club on the day of the game.
West Ham suffered more misery when Winston Reid appeared to land awkwardly in the first half, the central defender carefully treated for 10 minutes before leaving the field on a stretcher.
South Korea midfielder Ki Sung-Yueng drove a 20-yarder past West Ham goalkeeper Adrian in the eighth minute for only his second goal of the season.
The Foxes fell behind in the 35th minute at the King Power Stadium when Marc Albrighton conceded a penalty with a foul on Josh King, who picked himself up to convert the spot-kick.
And they didn't have to wait long to make it 2-0 as a corner by Ki saw Mike Van der Hoorn get in front of Rice and head into the net.
King, who was making his first start since joining on loan from Leicester, poked in a third which took Swansea above West Ham in the table.
West Ham's supporters have called off a protest march, which was due to take place before next weekend's home game against Burnley.
Under contract until 2020, there have been reports that West Ham will be offering Lanzini a new and improved contract but David Moyes has confirmed that the club will not be opening talks until their Premier League status is secured.
"West Ham also lost 4-1 last time out and their form has been mixed under David Moyes".
Swansea had not had a penalty in the Premier League since December 2016, but Jordan Ayew coolly converted his 10th goal of the season.
Both of West Ham's away victories this season have been by a three-goal margin: 3-0 at Stoke and 4-1 at Huddersfield.